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EllipticK






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticK[z] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/08.02.06.0019.01









  


  










Input Form





EllipticK[z] == (1/(2 Pi)) Sum[(1/k!) (-2 Pi I (-1)^k Floor[(Pi + Arg[1 - Subscript[z, 0]])/(2 Pi)] Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] Gamma[1/2 + k]^2 Hypergeometric2F1Regularized[1/2 + k, 1/2 + k, 1 + k, 1 - Subscript[z, 0]] + MeijerG[{{1/2 - k, 1/2 - k}, {}}, {{0, -k}, {}}, 1 - Subscript[z, 0]]) (z - Subscript[z, 0])^k, {k, 0, Infinity}]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticK", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]], "2"], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["1", "+", "k"]], ",", RowBox[List["1", "-", SubscriptBox["zz", "0"]]]]], "]"]]]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "k"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["1", "-", SubscriptBox["zz", "0"]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], RowBox[List["2", " ", "\[Pi]"]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02